(12/25/2010, 07:06 PM)sheldonison Wrote: I believe this is a graph of . So the line with c=0.5 would be the half iterate of the exp(z), which can be calculated as . For integer values of c, the equations are simpler. . And

Sorry, I should've been more specific. I meant to ask for a formula independent of tetration; I'm assuming there's a power series of some kind defining 0<= q<=1 Something that would reproduce this graph, since the linear model of tetration I've been using doesn't match up. Even a generalized power series for tetration would work, now that I think of it.

(12/25/2010, 07:53 PM)JmsNxn Wrote: Sorry, I should've been more specific. I meant to ask for a formula independent of tetration; I'm assuming there's a power series of some kind defining 0<= q<=1 Something that would reproduce this graph, since the linear model of tetration I've been using doesn't match up. Even a generalized power series for tetration would work, now that I think of it.

For base e exponent, as far as I know, the only way to generate the half iterate (or any other partial iterate) is indirectly, via . I'm unaware of any other way to generate the power series.
- Shel

(12/25/2010, 10:52 PM)sheldonison Wrote: For base e exponent, as far as I know, the only way to generate the half iterate (or any other partial iterate) is indirectly, via . I'm unaware of any other way to generate the power series.
- Shel

Sorry, you misunderstood me again. I meant precisely, does anybody know how to reproduce this graph? I have no means of evaluating ; although I know this is the derivation. I only know how to evaluate the linear approximation of tetration.

(12/26/2010, 03:33 AM)JmsNxn Wrote: Sorry, you misunderstood me again. I meant precisely, does anybody know how to reproduce this graph? I have no means of evaluating ; although I know this is the derivation. I only know how to evaluate the linear approximation of tetration.

Here are the sexp/slog Taylor series, for base e. Also, I updated my kneser.gp program which is at this link, http://math.eretrandre.org/tetrationforu...61#pid5461, which generates these Taylor series. You can also use kneser.gp to calculate sexp(slog(z)+0.5) directly.
- Sheldon

(12/25/2010, 10:52 PM)sheldonison Wrote: For base e exponent, as far as I know, the only way to generate the half iterate (or any other partial iterate) is indirectly, via . I'm unaware of any other way to generate the power series.
- Shel

you could also use my method since e > e^(1/2).

although its not directly a power series nor the best numerically method its simple and doesnt use slog or sexp.

(12/26/2010, 03:33 AM)JmsNxn Wrote: Sorry, you misunderstood me again. I meant precisely, does anybody know how to reproduce this graph? I have no means of evaluating ; although I know this is the derivation. I only know how to evaluate the linear approximation of tetration.

Here are the sexp/slog Taylor series, for base e. Also, I updated my kneser.gp program which is at this link, http://math.eretrandre.org/tetrationforu...61#pid5461, which generates these Taylor series. You can also use kneser.gp to calculate sexp(slog(z)+0.5) directly.
- Sheldon

(12/29/2010, 10:15 PM)JmsNxn Wrote: Thank you. Just one final question, how did you generate this Taylor series? Is there a closed form expression?

James, as far as I see this is a graph which was produced by Dimitri Kousnetzov, who also posted here in the forum (you may use this link to find all all posts of him ) and to a certain extend explained his method here. But there is also a published paper of him where he describes this in detail (I've never understood it, btw, because I seem to lack some basic knowledge about cauchy-integrals and riemann-mappings, but for a student of mathematics this may be completely familiar). I think his article is also in our (the tetration-forum's) database of literature (see the related message lit-ref-db in the forum)

The basic idea using base e here, where L is the fixed point such that , , is that if and and etc.

This can be used to develop a complex valued entire superfunction such that for all values of z.
The problem is that the superf is complex valued, not real valued. A 1-cyclic mapping is used to convert this function to an analytic real valued tetration. The 1-cyclic theta mapping is equivalent to the Riemann mapping in Kneser's algorithm, although convergence is not proven.http://math.eretrandre.org/tetrationforu...hp?tid=487

The Taylor series is generated via a unit circle Cauchy integral.
- Sheldon